# Signals and Systems â€“ Relation between Convolution and Correlation

## Convolution

The convolution is a mathematical operation for combining two signals to form a third signal. In other words, the convolution is a mathematical way which is used to express the relation between the input and output characteristics of an LTI system.

Mathematically, the convolution of two signals is given by,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( t-\tau \right )d\tau =\int_{-\infty }^{\infty }x_{2}\left ( \tau \right )x_{1}\left ( t-\tau \right )d\tau}$$

## Correlation

The correlation is defined as the measure of similarity between two signals or functions or waveforms. The correlation is of two types viz. cross-correlation and autocorrelation.

The cross correlation between two complex signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(đť‘ˇ) is given by,

$$\mathrm{R_{12}\left ( \tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( t \right )x^{\ast }_{2}\left ( t-\tau \right )dt =\int_{-\infty }^{\infty }x_{1}\left ( t+\tau \right )x^{*}_{2}\left ( t \right )dt}$$

If đť‘Ą1(đť‘ˇ) and đť‘Ą2(đť‘ˇ) are real signals, then,

$$\mathrm{R_{12}\left ( \tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( t \right )x_{2}\left ( t-\tau \right )dt =\int_{-\infty }^{\infty }x_{1}\left ( t+\tau \right )x_{2}\left ( t \right )dt}$$

## Relation between Convolution and Correlation

The convolution and correlation are closely related. In order to obtain the crosscorrelation of two real signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(đť‘ˇ), we multiply the signal đť‘Ą1(đť‘ˇ) with function đť‘Ą2(đť‘ˇ) displaced by τ units. Then, the area under the product curve is the cross correlation between the signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(đť‘ˇ) at đť‘ˇ = đťśŹ.

On the other hand, the convolution of signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(đť‘ˇ) at đť‘ˇ = đťśŹ is obtained by folding the function đť‘Ą2(đť‘ˇ) backward about the vertical axis at the origin [i.e., đť‘Ą2(−đť‘ˇ)] and then multiplied. The area under the product curve is the convolution of the signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(đť‘ˇ) at đť‘ˇ = đťśŹ.

Therefore, it follows that the correlation of signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(đť‘ˇ) is the same as the convolution of signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(−đť‘ˇ).

## Analytical Explanation

The resemblance between the convolution and the correlation can be proved analytically as follows −

The convolution of two signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(−đť‘ˇ) is given by,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( \tau-t \right )d\tau\: \: \: \cdot \cdot \cdot \left ( 1 \right )}$$

By replacing the variable đťśŹ by another variable p in the integral of Eqn. (1), we get,

$$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )=\int_{-\infty }^{\infty }x_{1}\left ( p \right )x_{2}\left ( p-t \right )dp\: \: \: \cdot \cdot \cdot \left ( 2 \right )}$$

Now, replacing the variable đť‘ˇ by đťśŹ in Eqn. (2), we have,

$$\mathrm{x_{1}\left ( \tau \right )\ast x_{2}\left ( -\tau \right )=\int_{-\infty }^{\infty }x_{1}\left ( p \right )x_{2}\left ( p-\tau \right )dp=R_{12}\left ( \tau \right )}$$

Therefore, the relation between correlation and convolution of two signals is given by,

$$\mathrm{R_{12}\left ( \tau \right )=x_{1}\left ( t \right )\ast x_{2}\left ( -t \right )|_{t=\tau }}$$

Similarly,

$$\mathrm{R_{21}\left ( \tau \right )=x_{2}\left ( t \right )\ast x_{1}\left ( -t \right )|_{t=\tau }}$$

Hence, it proves that the correlation of signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(đť‘ˇ) is equivalent to the convolution of signals đť‘Ą1(đť‘ˇ) and đť‘Ą2(−đť‘ˇ).

Therefore, all the techniques used to evaluate the convolution of two signals can also be applied to find the correlation of the signals directly. Similarly, all the results obtained for the convolution are applicable to the correlation.

Note – If one of the signals is an even signal, let the signal đť‘Ą2(đť‘ˇ) is an even signal [i.e. đť‘Ą2(đť‘ˇ) = đť‘Ą2(−đť‘ˇ)]. Then, the cross-correlation and the convolution of the two signals are equivalent.